Why should you switch doors in the Monty Hall problem?

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1 / 8 Three doors and one carA car is behind one door and goats are behind the other two. Nobody on stage knows more than you yet, so each door has one chance in three. Tap a door to play, or follow these steps.

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01
Three doors and one car
A car is behind one door and goats are behind the other two. Nobody on stage knows more than you yet, so each door has one chance in three. Tap a door to play, or follow these steps.
02
Your pick: one chance in three
You pick door 1. The tag over it reads 1/3, and the other two doors together hold the remaining 2/3. Keep that 2/3 in mind; it does not disappear when a door opens.
03
The host opens a goat door
The host knows where the car is and must open a door with a goat. Door 2 hides the car, so he has no choice and opens door 3. If you had picked the car, he could have opened either.
04
Switching wins two games in three
Your door was right in 1 game of 3 when you picked it, and the host's move cannot change that. The other 2 games of 3, the car is behind the door he left shut. Press a button to switch or stay.
05
A hundred doors
With 100 doors you pick one and the host opens 98 goat doors, skipping one. Your door is still 1 in 100. The door he went out of his way to skip holds the car 99 times in 100.
06
Ten thousand games
The towers show 10,000 games played both ways on the same doors. Switching settles near 66.7% and staying near 33.3%; the white rings mark the exact values.
07
A host who does not know
Now the host opens a door at random and happens to show a goat. Games where he would have shown the car are thrown out, and in the games left switching and staying win equally, 1/2 each.
08
A host with a favourite door
This host always leaves door 2 shut when he can. Here he left door 3 shut instead, which he only does when the car is there. Switching wins for certain; the Bayes table in Controls shows why.

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